Abstract

This paper is devoted to a stochastic retarded reaction-diffusion equation on all -dimensional space with additive white noise. We first show that the stochastic retarded reaction-diffusion equation generates a random dynamical system by transforming this stochastic equation into a random one through a tempered stationary random homeomorphism. Then, we establish the existence of a random attractor for the random equation. And the existence of a random attractor for the stochastic equation follows from the conjugation relation between two random dynamical systems. The pullback asymptotic compactness is proved by uniform estimates on solutions for large space and time variables. These estimates are obtained by a cut-off technique.

1. Introduction

The study of stochastic functional differential equations is motivated by the fact that, when one wants to model some evolution phenomena arising in physics, chemistry, biology, and other sciences, some hereditary characteristics such as aftereffect, time-lag, and time delay can appear in the variables. On the other hand, one of the most interesting problems concerning stochastic functional differential equations is to understood the asymptotic behavior of the solutions when time grows to infinite, since it can provide useful information about the future of the phenomenon described in the model.

It is known that the asymptotic behavior of random systems can be captured by a random attractor, which was introduced in [1, 2] as an extension of the attractor theory of deterministic systems in [3–5]. For stochastic PDEs without any hereditary features, the existence of random attractors has been investigated by many authors; see, for example, [6–20] and the references therein. However, this problem is not well studied in the case of stochastic retarded PDEs.

In this paper, we investigate the asymptotic behavior of solutions to the following stochastic retarded reaction-diffusion equation with additive noise defined in the entire space :
where is a positive constant, is a given function defined on , is a nonlinear functional satisfying certain conditions, and is a two-sided infinite dimensional Wiener process on a probability space which will be specified later.

We note that the asymptotic behavior of several deterministic retarded PDEs on bounded domains was studied in [21–25], and the case of retarded Navier-Stokes equations on some unbounded domains was treated in [26]. The random attractor for retarded stochastic differential equations was considered in [27] by monotone methods. Recently, in the case of stochastic retarded lattice dynamical systems defined on the entire integer set, the existence of a random attractor was proved in [28, 29]. Here we prove the existence of a random attractor for the stochastic retarded reaction-diffusion equation defined in . It is worth mentioning that the asymptotic behavior of the nonretarded version of (1) was investigated recently in [9].

Notice that Sobolev embeddings are not compact when domains are unbounded. This introduces a major obstacle for proving existence of attractors for PDEs on unbounded domains. Under certain circumstances, the tail-estimates method can be used to deal with the problem caused by the unboundedness of domains. This approach was developed in [30, 31] for deterministic nonretarded PDEs and used in [9–11, 13, 17, 18] for stochastic systems. At the same time, the present of delays is another obstacle, which makes phase spaces not reflexive and increases the difficulty of uniform estimates. In this paper, we will develop a tail-estimates approach for stochastic retarded PDEs on unbounded domains and prove the existence of a compact random attractor for the stochastic retarded reaction-diffusion equation (1), in particular, defined on the unbounded domain . The idea is based on the observation that the solutions of the equation are uniformly small when space and time variables are sufficiently large. It is clear that our method can be used for a variety of other equations, as it was for the nonretarded case.

For convenience, hereafter we adopt the following notations. We denote by and the norm and the inner product in . Otherwise, the norm of a general Banach space is written as . For , let denote the Banach space of all continuous functions endowed with the supremum norm . For any real number , , and any continuous function , denotes the element of given by for .

The rest of the paper is organized as follows. In the next section, we introduce basic concepts concerning random dynamical systems and random attractors. In Section 3, we define a continuous random dynamical system for the stochastic retarded reaction-diffusion equation on . The existence of the random attractor is given in Section 4.

2. Preliminaries

In this section, we introduce some basic concepts related to random attractors for random dynamical systems. The reader is referred to [1, 2, 6, 32–34] for more details.

Let be a separable Banach space with Borel -algebra and a probability space.

Definition 1. is called a metric dynamical system if is -measurable, and is the identity on , for all and for all .

Definition 2. A set is called invariant with respect to , if, for all , it holds that

Definition 3. A continuous random dynamical system on over a metric dynamical system is a mapping
which is -measurable, and, for all ,(i) is continuous for all ; (ii) is the identity on ; (iii) for all .

Definition 4. A random set is a multivalued mapping such that, for every , the mapping is measurable, where is the distance between the element and the set . It is said that the random set is bounded (resp., closed or compact) if is bounded (resp., closed or compact) for a.e. .

Definition 5. A random variable is called tempered with respect to , if for a.e.
A random set is called tempered if is contained in a ball with center zero and tempered radius for all .

Remark 6. If is tempered, then(1) for any , is tempered; (2)for any and a.e. and is tempered.

Moreover, if, for a.e. , is continuous in , then is continuous in for such , and, for any , is tempered.

Hereafter, we always assume that is a continuous random dynamical system over , and is a collection of random subsets of .

Definition 7. A random set is called a random absorbing set in if, for every and a.e. , there exists such that

Definition 8. A random set is called a -random attractor (-pullback attractor) for if the following hold: (i) is a random compact set; (ii) is strictly invariant; that is, for a.e. and all ,
(iii) attracts all sets in ; that is, for all and a.e. ,
where is the Hausdorff semimetric given by for any and .We remark that if , then this attractor is unique [33].

Definition 9. is said to be -pullback asymptotically compact in if, for all and a.e. , has a convergent subsequence in whenever , and .

The following existence result on a random attractor for a continuous random dynamical system can be found in [2, 34]. First, recall that a collection of random subsets of is called inclusion closed if whenever is an arbitrary random set and is in with for a.e. , then must belong to .

Proposition 10. Let be an inclusion-closed collection of random subsets of and a continuous random dynamical system on over . Suppose that is a closed random absorbing set for in and is -pullback asymptotically compact in . Then has a unique -random attractor which is given by

In this paper, we will take as the collection of all tempered random subsets of and prove the stochastic retarded reaction-diffusion equation on has a -random attractor.

In this section, we show that there is a continuous random dynamical system generated by the stochastic retarded reaction-diffusion equation on with additive white noise:
with the initial condition
Here is a positive constant, is a given function in , is an -valued two-sided Wiener process with a symmetric nonnegative finite trace covariance operator defined on a probability space which will be specified below, and is a continuous mapping satisfying the following conditions: ; there exists a positive continuous function with
for some positive integer such that, for all , with and ,
there exist positive constants and such that, for all , , , and ,
.

In the sequel, we consider the probability space where
is the Borel -algebra induced by the compact-open topology of , and the corresponding Wiener measure on with respect to the covariance operator . Let
Then is an ergodic metric dynamical system. Since the above probability space is canonical, we have
Similar to Proposition A.1 in [34], we can find that there exists a full -measure -invariant set such that for each

Let be the -completion of , and let
with
where is the smallest -algebra generated by the random variable for all , such that and is the collection of -null sets of .

Note that
so is a filtered metric dynamical system (see [32, pages 72 and 91] for more details). In addition, it is important to note that the measurability of is not true if we replace by its completion; see [32, page 547] for details.

In this paper, the solution of problem (10)-(11) is interpreted in a mild sense:
a.s. for any , where is the analytic semigroup on generated by . By the theory in [35], we deal with (22) on the complete probability space .

We now associate a continuous random dynamical system with the stochastic retarded reaction-diffusion equation (10)-(11) over . To this end, we introduce an auxiliary Ornstein-Uhlenbeck process on and transform the stochastic retarded reaction-diffusion equation into a random one. Let
Then by (18), (23) is well defined. The process , , is a stationary, Gaussian process. By Lemma 5.13 in [35], we can see that it is a mild solution of the linear equation
That is, for all and a.s.
Moreover, the random variable is tempered, and, for each , the mapping is continuous.

Setting for in (22), then by (25), we obtain a deterministic equation, a.s. in ,
which is the mild form of the evolution equation
with the initial condition
Here , .

Problem (27)-(28) is a deterministic partial functional differential equation with random coefficients, which can be solved pathwise. We now establish the following result for problem (27)-(28).

Theorem 11. Let and be fixed. Then the following properties hold:(1) for each , problem (27)-(28) has a unique mild solution that belongs to
for any , and for a.e. (2) Let and be the mild solutions of problem (27)-(28) for the initial data and , respectively. Then there exists a constant such that for all

Proof. (1) By -, following the same lines of Theorem in [36], one can show that, for each , there exists a such that (26) has a unique solution on . Moreover, if , then
We prove now that this local solution is a global one. For fixed , by regularity of mild solutions for an analytic semigroup [37, page 145], we inform that
for any , and (27) holds for a.e. . Then, taking the inner product of (27) with in , we get that
By (), we can choose small enough such that . Using the Young inequality, we find that
Then it follows from (34) and (35) that
Choose small enough such that . Then by (36), we obtain
Now, we can also choose such that . Integrating (37) over leads to
Using the Young inequality and , we find that
Then by (38) and (39), we obtain
where . Consequently,
Hence, for fixed , we get that, for ,
and, for ,
In view of (42) and (43), we find that, for all ,
Therefore, for all ,
which, together with (32), implies that . This proves property (1).(2) By (44), there exists a constant such that
Then from (A2) and (26), we have that for
Hence, for fixed , we get that, for ,
and, for ,
In view of (48) and (49), we find that, for all ,
The Gronwall inequality implies that, for all ,
This prove property (2). The proof is complete.

Conversely, if, for each , is a mild solution of problem (27)-(28) with , then by (25) the continuous process
is a mild solution of problem (10)-(11).

Theorem 12. Problem (27)-(28) generates a continuous random dynamical system over , where
Moreover, if one defines by
then is another continuous random dynamical system associated with problem (10)-(11).

Proof. By a classical successive approximation argument, one can easily show that, for fixed , is an -adapted continuous process. Hence, for fixed , is also an -adapted continuous process. On the other hand, from property (2) of Theorem 11, it follows that, for fixed and , is continuous. Consequently, is -measurable.By (26) we have that, for , and ,
Then again by (26) we get
For each consider
Then for we have
It follows from (56) that
for all . By the uniqueness of the solution of (26) we find that
while (58) implies
Therefore, is a continuous random dynamical system.As for , noticing that
we get from (61) that, for ,
Therefore, is also a continuous random dynamical system. Furthermore, and are conjugated random dynamical systems; that is,
where, for every , is a homeomorphism of . The proof is complete.

4. Existence of Random Attractors

In this section, we prove the existence of a -random attractor for the random dynamical system associated with the stochastic retarded reaction-diffusion equation (10)-(11) on . We first establish the existence of a -random attractor for its conjugated random dynamical system , then the existence of a -random attractor for follows from the conjugation relation between and . To this end, we will derive uniform estimates on the mild solutions of problem (27)-(28) when with the purpose of proving the existence of a bounded random absorbing set and the asymptotic compactness for . In particular, we will show that the tails of the solutions, that is, solutions evaluated at large value of , are uniformly small when time is sufficiently large.

From now on, we always assume that is the collection of all tempered subsets of with respect to . The next lemma shows that has a random absorbing set in .

Lemma 13. There exists such that is a random absorbing set for in ; that is, for any and a.e. , there exists such that

Proof. Replacing with in (41) and (44), we get that, for all ,
By assumption, is tempered. On the other hand, by Remark 6, is also tempered. Therefore, if , then there exists such that, for all ,
where
is tempered by Remark 6. Then it follows from (66), (67), and (68) that for all ,
Given , we define
where
is tempered. Then . Further, (71) indicates that is a random absorbing set for in , which completes the proof.

We next derive uniform estimates for in . From property (1) of Theorem 11 and the fact that [37, page 165]
for any , we get that .

Lemma 14. Let and . Then for a.e. , the solution of problem (27)-(28) satisfies, for all ,
where and are tempered and is the random function in Lemma 13.

Proof. Replacing with in (44), we get that, for all and ,
Integrating (76) over the interval leads to
which, together with (68), implies that, for all ,
Obviously, is tempered. Integrating (37) over the interval leads to
Using the Young inequality and , we obtain that
It follows from (79) and (80) that
Thus,
Replacing with in (82), we get from (71) that, for a.e. and all ,
Therefore, we have that, for all ,
By Remark 6, is tempered. Then the lemma follows from (78) and (84).

Lemma 15. Let and . Then for a.e. , the solution of problem (27)-(28) satisfies, for all and ,
where and are tempered and is the random function in Lemma 13.

Proof. Taking the inner product of (27) with in , we get that
Using the Young inequality, we obtain that
It follows from (86) and (87) that
Thus,
Let be the positive constant in Lemma 13, and take and . Integrating (89) over the interval leads to
Integrating the above with respect to over the interval , we get that
Replacing with and by - and Lemmas 13 and 14, we find that, for all ,
where
is tempered by Remark 6. Then we have that, for all ,
By , one can easily see that is also tempered for any tempered random variable . Hence, is tempered.Let , . Integrating (89) over the interval leads to
Replacing with in (95) and by -, Lemma 13, and (94), we find that, for all and ,
where
is tempered by Remark 6. Similar to , is also tempered. Then the lemma follows from (94) and (96).

Lemma 16. Let and . Then for every and a.e. , there exist and such that the solution of problem (27)-(28) satisfies, for all ,

Proof. Let be a smooth function defined on such that for all , and
Then there exists a positive deterministic constant such that for all . Taking the inner product of (27) with in , we get that